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i had a question. namely, in the continuity equation we see that [tex] \frac{\partial}{\partial t}\iiint_V \rho dV = -\iint_{S} \rho \vec{v} \cdot d\vec{S}[/tex] and we may use the divergence theorem to have: [tex] \frac{\partial}{\partial t}\iiint_V \rho dV = -\iiint_{V} \nabla \cdot \big( \rho \vec{v} \big) dV[/tex]

ultimately, we arrive at: [tex] \frac{\partial}{\partial t}\bigg( \rho \bigg) = -\nabla \cdot \big( \rho \vec{v} \big)[/tex]

my question is, at this point, how are we able to do the following two things:

1interchange ##\frac{\partial}{\partial t}## inside the volume integral?

2drop the volume integrals entirely?

i should say that the volume is arbitrary, and from what i remember, we have to do something like ##\lim_{V \to 0}## but i don't know the formal, mathematical procedure here.

can someone please help me out?

thanks!

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# When using stokes theorem to remove integrals

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